Finite Linear Spaces with a Regularity Condition on the Transversals of Two Secant Lines

Projective and affine planes are, besides the degenerate projective planes, the only known examples of finite linear spaces for which there is a non-negative integer t such that for any two intersecting lines L, L′ of S and any point p outside L ∪ L′, there are exactly t transversals of L and L′ passing through p. We prove that all finite linear spaces satisfying the above condition are either degenerate projective planes or Steiner systems S(2, k, v) in which t and k must satisfy some rather restrictive arithmetic conditions